A non-linear analysis of Turing pattern formation

Reaction-diffusion schemes are widely used to model and interpret phenomena in various fields. In that context, phenomena driven by Turing instabilities are particularly relevant to describe patterning in a number of biological processes. While the conditions that determine the appearance of Turing patterns and their wavelength can be easily obtained by a linear stability analysis, the estimation of pattern amplitudes requires cumbersome calculations due to non-linear terms. Here we introduce an expansion method that makes possible to obtain analytical, approximated, solutions of the pattern amplitudes. We check and illustrate the reliability of this methodology with results obtained from numerical simulations.